The cartesian product is a feature of euclidean geometry. You can apply it by drawing shapes in different sets of axies, and then taking the common set.

For a hexagonal prism, you could cut a length from a hexagonal bar. This gives a prism as an off-cut (prisma means offcut!). Alternately, you could take a slab of constant thickness, and cut a pentagon from it. In either way, all space -> layer -> pentagonal prism.

In four dimensions, there are four axies, and so prisms can be cut from 2+2 axies, or 3+1. The former gives for example, polygon-polygon "duoprisms", such as the triangle-triangle (duo)prism. Circles behave as polygons, and a special name is used for the bi-circular prism (duocylinder), rather like a special name is used for the circular prism in 3d (cylinder).

"Duopyramid" is confusing in four dimensions, since you can erect a pyramid on a pyramidal base, or (as in 3d), erect a pyramid on each side of a 3d base. In any case, the term is depreciated. The figure here has no special name over bi-triangular (duo)prism.

Pyramids do not occur in the rotopes.

You can indeed use powers to generate higher polytopes. The products are all discovered in just this way.

kingmaz wrote:As a start, I'll underline that I don't understand the mechanism of the Cartesian Product.

However, if the Cartesian Product of two circles is the duocylinder, does that mean the Cartesian Product of two triangles is a duopyramid?

The cartesian product of two triangles is a 3,3-duoprism. There is no such thing as a duopyramid.

Is this the same as Rotope 55?

Rotope 55 is 5D (a 3,3-duoprism is 4D). R55 is a square pyramid prism pyramid which has nothing to do with a duoprism.

Moving on from that, what is Cartesian Product of two pyramids called (Rotope 188?)?

See my response to the next quote. Rotope 188 is not the cartesian product of two pyramids, it's a square pyramid diprism pyramid.

In general superscripts in rotopic group notation DO NOT imply Cartesian products; you seem to be assuming they do.

I suggest staying away from rotopic group notation as as I've said many times it's a huge mess. Use SSC2 instead.

Do the 'Cartesian Squares' (in effect) follow a regular pattern or do each have to be considered on an individual basis?

Yes, cartesian squares are indeed powertopes. The cartesian product of X and X can be called the Xar tetragoltriate. So your cartesian product of two pyramids (I'll assume square pyramids) would be called a square pyramidal tetragoltriate. Note that a duocylinder could be called a circular tetragoltriate and a tesseract could be called a square tetragoltriate.

This would lead onto a question regarding the use of indices/powers in generating higher dimensional figures, but I'll leave that for the moment...

Thanks for the replies both. I do appreciate you taking the time to correct me. I'm very glad that the things that crop up in my mind have been dealt with in advance and you can explain where I'm going wrong in certain thoughts and assumptions. Still it's early days for me in this subject.

I'll take the point that rotopic notation is outdated though initially quite easy looking, it's just that SSC2 seems a bit like german - difficult to get used to the rules, but it's very regular and straightforward once you're there.

The confusion over Rotope 55 arose due to me reading 11'1' as 1'1', apologies.

"Rotope 188 is not the cartesian product of two pyramids, it's a square pyramid diprism pyramid."I was trying to use duopyramid for the fact that 11' followed by 11' appeared to be doing the same thing twice, rather than confusing ' with indices.For a very easy looking operation 11'11' (yes I should use SSC2, I know) this seems a very long name, I do prefer pyramidal tetragoltriate.Has an alternative naming system been proposed for successive tapered and linear extrusions?